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Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra

So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan. The problem is the following: Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
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1answer
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If operator is closed and densely defined then $D(A^*)^\perp = \{0\}$

I'm a bit rusty in my Functional Analysis and couldn't solve this question: Let $X$ be a Banach space (over either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Show that, if $A:D(A) \...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
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Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
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Finding a basis of the annihilator of a subspace

Let $V\subset \mathbb{R}^4$ be the subspace spanned by $e_1+e_2+e_3+e_4$ and $e_1+2e_2+3e_3+4e_4$. Find a basis of the subspace $V^{\circ}$ in the dual space $(\mathbb{R})^*$. My attempt: Let $a = ...
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Why are $T^k(V)$ and $V^* \otimes… \otimes V^*$ just isomorphic?

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* \otimes... \otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $\epsilon^1,.....
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Commutative diagrams for vector spaces, dual spaces, and adjoint of linear maps

I'm looking for a commutative diagram explaining the relationship between a vector space, $X$ the dual space, $X^*$, the double dual, $X^{**}$. I'm also looking for a diagram explaining the ...
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Value of linear function $\alpha(h)$ defined in dual basis

Let $V = \mathbb{R}[x]_{\leq 2}$ and $\{\varepsilon_1, \varepsilon_2, \varepsilon_3\}$ - dual basis in $V^*$ of basis $\{-1 - x -2x^2, 7 + 6 x + 13x^2, 9 + 15x + 23x^2\}$ in $V$. Also $\{f_1, f_2, ...
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Bra’s, Ket’s, a Hilbert Space and it’s Dual.

So I’m trying to get this all straightened out in my head. In Quantum mechanics we use a Hilbert Space $\mathcal{H}$ as our vector space and we say that its elements is the set of Ket’s $\left|\psi\...
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1answer
38 views

Construct a natural map $l_1 → l_{\infty}^∗$

I know from Show that $(l_1)^* \cong l_{\infty}$ that $l^∗_1$ is isometrically isomorphic to $l_{\infty}$. Indeed we can show that a map $L: l_{\infty} \rightarrow (l_1)^*$ given by $$L(x)(y) = \sum_{...
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If $T\in B(X,Y)$ and $T$ is bijective is $T^*$ also bijective?

1) This first link seems to provide a proof that seems to work for all normed vector spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? 2) This second ...
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Coordinates of the linear function in dual space. [closed]

Assume we have dual basis {$\phi_1,\phi_2, \phi_3 $} in $V^*$ of basis {$v_1, v_2, v_3$} in $\mathbb{R}^3$. How can i find coordinates of the function $f(x) = x_1 - x_2 + 4x_3$ in basis {$\phi_1, \...
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Unit closed Ball in completion

Let $(E, \|\cdot\|_E)$ be any normed vector space. Consider the canonical identification with its double dual $(E'', \|\cdot\|_{E''})$ given by $$i_E: (E, \|\cdot\|_E) \to (E'', \|\cdot\|_{E''}), \; \...
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1answer
31 views

Examples reflexive spaces

Let $(E, \| \cdot \|_E)$ be a normed vector space over a field $\mathbb{K}$ and $(E', \| \cdot \|_{\mathrm{op}})$ its dual. Theorem 1). If $(E, \| \cdot \|_E)$ is reflexive, then each bounded ...
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Is the identity functor naturally isomorphic to a covariant dual functor?

It is often said that vector spaces are not naturally isomorphic to dual spaces, because the dual functor is not naturally isomorphic to the identity functor. But the latter is a rather trivial ...
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Riesz representation theorem: Does the order matter?

Let $X$ be a Hilbert space. $J:X\rightarrow X',\hspace{1cm}J(x):=(\cdot,x)$ is a complex conjugated isometric isomorphism between $X$ and it's dual space $X'$. Would there be any problems as a ...
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Does the dual space of a topological vector space always separate points? [duplicate]

If $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, separates points. What that means is that if $x_1,x_2\in X$, then there exists an $f\in ...
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1answer
39 views

Hahn-Banach needed to show equality?

Let $X$ be a normed space and $x_1, x_2 \in X$. Suppose $x^{\ast}(x_1) = x^{\ast}(x_2)$ for all $x^{\ast} \in X^{\ast}$. Then $x_1 = x_2$. Do we need Hahn-Banach (hence, equivalently some sort of ...
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$C(X)$ not reflexive if $X$ has infinitely many points.

Let $X$ be a compact metric space with infinitely many points, then show $C(X)$ is not reflexive. I think I see why this is the case for $X=[a,b]$, but I don't see how one can extend it. Is there a ...
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If $Y\subset X$ are Banach spaces such that $Y$ is dense in $X$, is it true that $X'$ is dense in $Y'$?

If $Y$ is a dense subspace of a Banach space $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ is a Banach space such that the inclusion from $(Y,\|\cdot\|_2)$ into $(X,\|\cdot\|_1)$ is continuous, then it is ...
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How can the adjoint be defined if $f$ is not one to one

Let $f \in \mathcal{L}(V, W)$. Moreover let's suppose $(e_1, ..., e_n)$ is a basis of $V$ and $f(e_i) = v_i$ where the $v_i$ aren't distinct (so there is at least $i \ne j$ such that $v_i = v_j$) so ...
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1answer
47 views

Why the transpose of $A$ is not looking at the transformation $A$ in the dual space

Most of the time people are saying that in order to have a better understanding about what the transpose really represents is this: We have a linear transformation $T : V \to V$ with $V$ a finite ...
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1answer
27 views

Why is the annilihator of the zero linear functional the vector space itself?

Let V be a vector space of a finite dimension. $ T:V \rightarrow V $ is a linear transformation. I have to prove that T* is injective iff T is injective. I know T* is injective iff $kerT^* = 0$, and $...
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2answers
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Intuition: Dual Space is always Banach

Theorem. Let $X$ be a normed space and $Y$ be a Banach space. Then the set of continuous linear maps $L(X,Y)$ is a Banach space (with the operator norm). From the well-known theorem above, we get an ...
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Constructing the Algebraic Dual Space as a K-Algebra

Fix an arbitrary Field $K$ and suppose we are given a Vector Space $V\in{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$. It's clear that $[V\in{Obj(Vect_{...
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1answer
52 views

Isomorphism between tensor product of vector fields and their dual.

Consider two finite dimensional vector spaces $V_1,V_2$ and their duals denoted by $V_1^{*},V_{2}^{*}$. I am working on a problem that is asking me to prove a generalized version of the below, but I ...
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4answers
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$X'$ finite-dimensional implies $X$ finite-dimensional

How would one prove, for any normed space $X$ that if $X'$ is finite dimensional, then $X$ is finite-dimensional? Here $X'$ denotes the space of all bounded functionals $f: X \to\mathbb F$ If anyone ...
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Dunford-Pettis for Banach-Spaces

Can anyone tell me if the Dunford-Pettis property is met for a separate refelxive Banach space $X$ with dual $X'$? I would say that this is the case.
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1answer
19 views

Image unit ball under Isometric Isomorphism

Let $X,Y$ be a normed spaces, not necessarily finitely dimensional and let $T\in B(X,Y)$ be an isometric isomorphism. I want to show the same holds for the dual of $T$, $T'\in B(Y',X')$. The ...
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1answer
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Non-reflexive space that is isomorphic to its second dual space

I was wondering if it is possible to construct a space that is non-reflexive (so it is not isomorphic to its second dual space under the cannonical embedding), but some isomorphism exists between them....
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Banach spaces and isomorphism between dual spaces

Maybe is a silly question, but I have got a doubt about it: Let $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ be two linearly isomorphic Banach spaces. Then we know that their dual spaces are ...
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3answers
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Finding Dual Basis from a basis in $\mathbb R^2$

I am given the fact that these two vectors form the basis B of $\mathbb R^2$: $$ B=\{\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} \begin{bmatrix} 3 \\ 1 \end{bmatrix}\} $$ and then asked to find the dual ...
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1answer
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Question on Banach-Alaoglu theorem: Bounded subset of a set contained in the dual space

So the Banach-Alaoglu theorem states: Let $X$ be the dual space to some Banach separable space $Z$, i.e $X=Z^*$. Take $M$ a bounded subset of $X$. Then any sequence in $M$ has a weak-* ...
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1answer
37 views

Dualization map is surjective

I am practicing for my exam and I want to solve the following problem. Let $X,Y$ be normed reflexive spaces. Show that the "Dualization map" $':B(X,Y)\to B(Y',X')$, $T\mapsto T'$ is surjective I ...
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2answers
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Dualilty map in Banach space

Let $X$ be a Banach space, we define $ \phi_-(y)=\lim_{t\rightarrow{0^+}}=\frac{|x|-|x-ty|}{t} $ $\phi_+(y)=\lim_{t\rightarrow{0^+}}=\frac{|x+ty|-|x|}{t} $ Then $ M^*(x)= \{ x* \in X^*: \phi_-(y)\...
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1answer
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Eigenvalue of an operator implies eigenvalue of the dual?

I helped some students today with linear algebra, which I took last year. They asked me a question from their homework to which I couldn't find an answer: Let $V$ be a finitely generated vector space ...
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1answer
29 views

Can you take the dot product of a column vector and a row vector (i.e. a vector and a dual vector)

I recently learned about the definition of work, namely the one involving path integration. W=integral of (F.dr). In this case, F is a vector field and dr is a small segment of the path r being ...
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19 views

Show that $A^o$ is convex, balanced, closed in $A'$

Let $E$ is a normed space and $A \subset E$. Define $$A^o := \{y \in A': |y(x)| \leq 1, \forall x \in A\}.$$ in which, $A'$ is the dual space of $A$. With this definition, how to show that $A^o$ is ...
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1answer
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$B$ Banach $\implies B^*$ Banach.

Since we only need to check that $B^*$ is complete, we should prove Cauchy sequence $\lbrace l_n \rbrace$ converges. In general idea, $$|(l-l_n)(f)| \leq |(l-l_m)(f)| + |(l_m-l_n)(f)| \leq |(l-l_m)(f)...
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surjectivity of dual operator congruence implies reflexivity

I tried to prove the following: Let $X,Y$ be normed spaces and $\Phi:\mathcal{L}(X,Y)\to\mathcal{L}(Y^*,X^*)$, $\Phi(A)=A^*$ is surjective. Then $Y$ is reflexive. Here $X^*$ $(A^*)$ denotes the dual ...
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3answers
89 views

Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective

I am working on the dual spaces of sequence spaces, and I want to show that the map $$ \Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i $$ is not surjective. I have already ...
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1answer
26 views

When are the topological duals $A^*$ and $B^*$ isomorphic?

If I have to normed vector spaces $A$ and $B$, I was wondering when the topological duals are isomorphic (i.e. $A^* \cong B^*)$ . Is it sufficient that $A \cong B$? Or that $A$ has to be isometric to $...
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1answer
37 views

Duality: dual maps and linear maps

Suppose we have $V$ and $W$ be $2$ $K$-vectorspaces and $f:V\rightarrow W$ is a linear map then: For all $\varphi\in W^\ast$ one has $\varphi\circ f\in V^*$ The map $f^\ast:W^\ast\rightarrow V^\ast:\...
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38 views

Dual map and solving linear equation

Let $f:V\to W$ be a linear transformation from a vector space $V$ to a vector space $W$. Suppose that $b\in W$ satisfies $\phi(b)=0$ for all $\phi\in\ker(f^*)$. Show that there exists $x\in V$ such ...
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Given a linear operator $T$ and a linear functional $\phi_n(x)=(T(x))(n)$, show that $T$ is continuous iff $\phi \in X^*$

Given $X$ a Banach space, and $T:X \rightarrow l_p$ a linear operator, with $1 \leq p \leq \infty$, for all $n \in \mathbb{N}$ consider the linear functional $\phi_n:X \rightarrow \mathbb{K}$, defined ...
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2answers
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Dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$.

I want to show that for $p\in(1,+\infty)$ the dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$ is isometrically isomorphic to $L^q(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$, where $\frac{1}{p}+\...
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intersection in a simplex

In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
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2answers
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Which vector spaces are algebraic dual spaces?

Let us say that a vector space $V$ is an algebraic dual space if there exist a vector space $U$ such that $V$ is isomorphic to $U^*$, the vector space of all linear maps from $U$ to the corresponding ...
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1answer
63 views

The relationship between vector space dualization and matrix transposition.

Let $\mathbb{F}$ be a field. The category of matrices $\mathbf{Mat}$ has $\mathbb{N}_0$ as class of objects and $\mathrm{hom}(n,m)=\mathbb{F}^{n\times m}$ for $n,m\in\mathbb{N}_0$, with $\mathrm{id}_n=...
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2answers
66 views

Finding $B^*$, the dual basis

Find a basis $B$ for $$V = \left\{ \left[ \begin{array}{cc} x\\ y\\ z \end{array} \right] \in \mathbb{R}^3 \vert x+y+z = 0\right\}$$ and then find $B^*$, the dual basis for $B$. The way we ...